Hurwitz series rings satisfying a zero divisor property

Abstract

In this paper, we study zero divisors in Hurwitz series rings and Hurwitz polynomial rings over general noncommutative rings. We first construct Armendariz rings that are not Armendariz of the Hurwitz series type and find various properties of (Hurwitz series) Armendariz rings. We show that for a semiprime Armendariz of Hurwitz series type (so reduced) ring $R$ with $a.c.c.$ on annihilator ideals, $HR$ (the Hurwitz series ring with coefficients over $R$) has finitely many minimal prime ideals, say $B_1, \ldots, B_m$ such that $B_1 \cdot \ldots \cdot B_m = 0$ and $B_i = HA_i$ for some minimal prime ideal $A_i$ of $R$ for all $i$, where $A_1, \ldots, A_m$ are all minimal prime ideals of $R$. Additionally, we construct various types of (Hurwitz series) Armendariz rings and demonstrate that the polynomial ring extension preserves the Armendarizness of the Hurwitz series as the Armendarizness.